TY - JOUR
T1 - Asymptotic decomposition for semilinear wave and equivariant wave map equations
AU - Jia, Hao
AU - Kenig, Carlos
N1 - Publisher Copyright:
© 2017 by Johns Hopkins University Press.
PY - 2017/12
Y1 - 2017/12
N2 - In this paper we give a unified proof to the soliton resolution conjecture along a sequence of times, for the semilinear focusing energy critical wave equations in the radial case and two dimensional equivariant wave map equations, including the four dimensional radial Yang Mills equation, without using outer energy type inequalities. Such inequalities have played a crucial role in previous works with similar results. Roughly speaking, we prove that along a sequence of times tn→T+ (the maximal time of existence), the solution decouples to a sum of rescaled solitons and a term vanishing in the energy space, plus a free radiation term in the global case or a regular part in the finite time blow up case. The main difficulty is that in general (for instance for the radial four dimensional Yang Mills case and the radial six dimensional semilinear wave case) we do not have a favorable outer energy inequality for the associated linear wave equations. Our main new input is the simultaneous use of two virial identities.
AB - In this paper we give a unified proof to the soliton resolution conjecture along a sequence of times, for the semilinear focusing energy critical wave equations in the radial case and two dimensional equivariant wave map equations, including the four dimensional radial Yang Mills equation, without using outer energy type inequalities. Such inequalities have played a crucial role in previous works with similar results. Roughly speaking, we prove that along a sequence of times tn→T+ (the maximal time of existence), the solution decouples to a sum of rescaled solitons and a term vanishing in the energy space, plus a free radiation term in the global case or a regular part in the finite time blow up case. The main difficulty is that in general (for instance for the radial four dimensional Yang Mills case and the radial six dimensional semilinear wave case) we do not have a favorable outer energy inequality for the associated linear wave equations. Our main new input is the simultaneous use of two virial identities.
UR - http://www.scopus.com/inward/record.url?scp=85034864172&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85034864172&partnerID=8YFLogxK
U2 - 10.1353/ajm.2017.0039
DO - 10.1353/ajm.2017.0039
M3 - Article
AN - SCOPUS:85034864172
SN - 0002-9327
VL - 139
SP - 1521
EP - 1603
JO - American Journal of Mathematics
JF - American Journal of Mathematics
IS - 6
ER -